An optimal regularity result for Kolmogorov equations and weak uniqueness for some critical SPDEs

Abstract

We show uniqueness in law for the critical SPDE

$$\mathit{d}{\mathit{X}}_{\mathit{t}}=\mathit{A}{\mathit{X}}_{\mathit{t}}\mathit{d}\mathit{t}+{(-\mathit{A})}^{1/2}\mathit{F}(\mathit{X}(\mathit{t}))\mathit{d}\mathit{t}+\mathit{d}{\mathit{W}}_{\mathit{t}},{\mathit{X}}_0=\mathit{x}\in \mathit{H},$$

where $\mathit{A}:\text{dom}(\mathit{A})\subset \mathit{H}\to \mathit{H}$ is a negative definite self-adjoint operator on a separable Hilbert space H having ${\mathit{A}}^{-1}$ of trace class and W is a cylindrical Wiener process on H. Here, $\mathit{F}:\mathit{H}\to \mathit{H}$ can be continuous with, at most, linear growth (some functions F which grow more than linearly can also be considered). This leads to new uniqueness results for generalized stochastic Burgers equations and for three-dimensional stochastic Cahn–Hilliard-type equations which have interesting applications. To get weak uniqueness, we use an infinite dimensional localization principle and also establish a new optimal regularity result for the Kolmogorov equation $\mathit{\lambda }\mathit{u}-\mathit{L}\mathit{u}=\mathit{f}$ associated to the SPDE when $\mathit{F}=0$ ($\mathit{\lambda }>0$, $\mathit{f}:\mathit{H}\to \mathbb{R}$ Borel and bounded). In particular, we prove that the first derivative $\mathit{D}\mathit{u}(\mathit{x})$ belongs to $\text{dom}({(-\mathit{A})}^{1/2})$, for any $\mathit{x}\in \mathit{H}$, and ${sup}_{\mathit{x}\in \mathit{H}}|{(-\mathit{A})}^{1/2}\mathit{D}\mathit{u}(\mathit{x}){|}_{\mathit{H}}=\Vert {(-\mathit{A})}^{1/2}\mathit{D}\mathit{u}{\Vert }_0\le \mathit{C}\Vert \mathit{f}{\Vert }_0$.